Superconformal algebras for twisted connected sums and G2 mirror symmetry JHEP 1812, 011 (2018) [1809.06376] Marc-Antoine Fiset University of Oxford KITP, Santa Barbara, 12 April 2019 (Md ; g()Mj Ricci(d ; g) g) ≈ 0 2-dim Σ2 (complex curve; coords z; z¯) σ :Σ2 ! Md In coords: x i (z; z¯), i = 1;::: d (Field) Principle to select trajectory σ (Theory) (Polyakov action / Non-linear sigma model) RNS Superstrings RNS Superstrings (IIA, IIB, heterotic): (IIA, IIB, heterotic): Quantization Add fermions Add spinors (Canonical / WZW / α0-expansion) 2-dim2-dimN = QFT 1 SQFTSCFT Introduction d d 1;9−d (M ; g) M × R Not M-theory! (Md ; g()Mj Ricci(d ; g) g) ≈ 0 2-dim Σ2 (complex curve; coords z; z¯) σ :Σ2 ! Md In coords: x i (z; z¯), i = 1;::: d (Field) Principle to select trajectory σ (Theory) (Polyakov action / Non-linear sigma model) RNS Superstrings RNS Superstrings (IIA, IIB, heterotic): (IIA, IIB, heterotic): Quantization Add fermions Add spinors (Canonical / WZW / α0-expansion) 2-dim2-dimN = QFT 1 SQFTSCFT Introduction d d 1;9−d (M ; g) M × R Not M-theory! (Md ; g()Mj Ricci(d ; g) g) ≈ 0 2-dim Σ2 (complex curve; coords z; z¯) σ :Σ2 ! Md In coords: x i (z; z¯), i = 1;::: d (Field) Principle to select trajectory σ (Theory) (Polyakov action / Non-linear sigma model) RNS Superstrings RNS Superstrings (IIA, IIB, heterotic): (IIA, IIB, heterotic): Quantization Add fermions Add spinors (Canonical / WZW / α0-expansion) 2-dim2-dimN = QFT 1 SQFTSCFT Introduction d d 1;9−d (M ; g) M × R Not M-theory! (Md ; g()Mj Ricci(d ; g) g) ≈ 0 σ :Σ2 ! Md In coords: x i (z; z¯), i = 1;::: d (Field) Principle to select trajectory σ (Theory) (Polyakov action / Non-linear sigma model) RNS Superstrings RNS Superstrings (IIA, IIB, heterotic): (IIA, IIB, heterotic): Quantization Add fermions Add spinors (Canonical / WZW / α0-expansion) 2-dim2-dimN = QFT 1 SQFTSCFT Introduction d d 1;9−d (M ; g) M × R Not M-theory! 2-dim Σ2 (complex curve; coords z; z¯) (Md ; g()Mj Ricci(d ; g) g) ≈ 0 Principle to select trajectory σ (Theory) (Polyakov action / Non-linear sigma model) RNS Superstrings RNS Superstrings (IIA, IIB, heterotic): (IIA, IIB, heterotic): Quantization Add fermions Add spinors (Canonical / WZW / α0-expansion) 2-dim2-dimN = QFT 1 SQFTSCFT Introduction d d 1;9−d (M ; g) M × R Not M-theory! 2-dim Σ2 (complex curve; coords z; z¯) σ :Σ2 ! Md In coords: x i (z; z¯), i = 1;::: d (Field) (Md ; g()Mj Ricci(d ; g) g) ≈ 0 RNS Superstrings RNS Superstrings (IIA, IIB, heterotic): (IIA, IIB, heterotic): Quantization Add fermions Add spinors (Canonical / WZW / α0-expansion) 2-dim2-dimN = QFT 1 SQFTSCFT Introduction d d 1;9−d (M ; g) M × R Not M-theory! 2-dim Σ2 (complex curve; coords z; z¯) σ :Σ2 ! Md In coords: x i (z; z¯), i = 1;::: d (Field) Principle to select trajectory σ (Theory) (Polyakov action / Non-linear sigma model) (Md ; g()Mj Ricci(d ; g) g) ≈ 0 RNS Superstrings RNS Superstrings (IIA, IIB, heterotic): (IIA, IIB, heterotic): Add fermions Add spinors 2-dim2-dimN = QFT 1 SQFTSCFT Introduction d d 1;9−d (M ; g) M × R Not M-theory! 2-dim Σ2 (complex curve; coords z; z¯) σ :Σ2 ! Md In coords: x i (z; z¯), i = 1;::: d (Field) Principle to select trajectory σ (Theory) (Polyakov action / Non-linear sigma model) Quantization (Canonical / WZW / α0-expansion) (Md ; g) j Ricci(g) ≈ 0 RNS Superstrings RNS Superstrings (IIA, IIB, heterotic): (IIA, IIB, heterotic): Add fermions Add spinors 2-dim N = 1 SQFTSCFT Introduction d d 1;9−d (M ; g) M × R Not M-theory! 2-dim Σ2 (complex curve; coords z; z¯) σ :Σ2 ! Md In coords: x i (z; z¯), i = 1;::: d (Field) Principle to select trajectory σ (Theory) (Polyakov action / Non-linear sigma model) Quantization (Canonical / WZW / α0-expansion) 2-dim QFT (Md ; g) j Ricci(g) ≈ 0 2-dim2-dimN = QFT 1 SCFT Introduction d d 1;9−d (M ; g) M × R Not M-theory! 2-dim Σ2 (complex curve; coords z; z¯) σ :Σ2 ! Md In coords: x i (z; z¯), i = 1;::: d (Field) Principle to select trajectory σ (Theory) (Polyakov action / Non-linear sigma model) RNS Superstrings RNS Superstrings (IIA, IIB, heterotic): (IIA, IIB, heterotic): Quantization Add fermions Add spinors (Canonical / WZW / α0-expansion) 2-dim N = 1 SQFT (Md ; g) 2-dim2-dimN = QFT 1 SQFT Introduction d d 1;9−d (M ; g) j Ricci(g) ≈ 0 M × R Not M-theory! 2-dim Σ2 (complex curve; coords z; z¯) σ :Σ2 ! Md In coords: x i (z; z¯), i = 1;::: d (Field) Principle to select trajectory σ (Theory) (Polyakov action / Non-linear sigma model) RNS Superstrings RNS Superstrings (IIA, IIB, heterotic): (IIA, IIB, heterotic): Quantization Add fermions Add spinors (Canonical / WZW / α0-expansion) 2-dim N = 1 SCFT • Not surjective: • Not injective: 9 other constructions of SCFTs Mirror symmetry, T-duality Mirror symmetry for TCS G2? [Braun, Del Zotto '17; '18] Landau-Ginzburg, Gepner, etc. Introduction (Md ; g) j Ricci(g) ≈ 0 2-dim N = 1 SCFT • Not injective: Mirror symmetry, T-duality Mirror symmetry for TCS G2? [Braun, Del Zotto '17; '18] Introduction (Md ; g) j Ricci(g) ≈ 0 • Not surjective: 9 other constructions of SCFTs Landau-Ginzburg, Gepner, etc. 2-dim N = 1 SCFT Introduction (Md ; g) j Ricci(g) ≈ 0 • Not surjective: • Not injective: 9 other constructions of SCFTs Mirror symmetry, T-duality Mirror symmetry for TCS G2? [Braun, Del Zotto '17; '18] Landau-Ginzburg, Gepner, etc. 2-dim N = 1 SCFT Operator algebras TCS Contents 1 Operator algebras Formalism Examples (various Md ) 2 TCS Algebras for TCS G2 mirror symmetry M.-A. Fiset 1809.06376 1-dimensional Z2-graded Lie algebra \Operator algebra" N =1 fGn ; Tngn2Z ; c Lie bracket Vir c := G 3 (z) ; T(2)(z) OPEs 2 c : central charge 0 0 0 0 c=2 2T (z ) @z T (z ) 1 3 T (z)T (z ) ∼ + + [Tm; Tn] = (m − n)Tm+n + (m − m)c (z − z0)4 (z − z0)2 z − z0 12 T (z)G(z0) ∼ ::: [Tm; Gn] = ::: 0 [Gm; Gn] = ::: G(z)G(z ) ∼ ::: Generators(weights) Operator algebras Formalism TCS Examples Operator algebras: formalism (Md ; g) j Ricci(g) ≈ 0 2-dim N = 1 SCFT Virasoro N = 1 acts on the Hilbert space. M.-A. Fiset 1809.06376 \Operator algebra" N =1 Vir c := G 3 (z) ; T(2)(z) OPEs 2 c=2 2T (z0) @0 T (z0) T (z)T (z0) ∼ + + z (z − z0)4 (z − z0)2 z − z0 T (z)G(z0) ∼ ::: G(z)G(z0) ∼ ::: Generators(weights) Operator algebras Formalism TCS Examples Operator algebras: formalism (Md ; g) j Ricci(g) ≈ 0 2-dim N = 1 SCFT Virasoro N = 1 acts on the Hilbert space. 1-dimensional Z2-graded Lie algebra fGn ; Tngn2 ; c Lie bracket Z c : central charge 1 3 [Tm; Tn] = (m − n)Tm+n + (m − m)c 12 [Tm; Gn] = ::: [Gm; Gn] = ::: M.-A. Fiset 1809.06376 Operator algebras Formalism TCS Examples Operator algebras: formalism (Md ; g) j Ricci(g) ≈ 0 2-dim N = 1 SCFT Virasoro N = 1 acts on the Hilbert space. 1-dimensional Z2-graded Lie algebra \Operator algebra" N =1 fGn ; Tngn2Z ; c Lie bracket Vir c := G 3 (z) ; T(2)(z) OPEs 2 c : central charge 0 0 0 0 c=2 2T (z ) @z T (z ) 1 3 T (z)T (z ) ∼ + + [Tm; Tn] = (m − n)Tm+n + (m − m)c (z − z0)4 (z − z0)2 z − z0 12 T (z)G(z0) ∼ ::: [Tm; Gn] = ::: 0 [Gm; Gn] = ::: G(z)G(z ) ∼ ::: Generators(weights) M.-A. Fiset 1809.06376 Operator algebras Formalism TCS Examples Operator algebras: formalism Assemble your favorite name! (closely related concepts): Chiral (Super-)Conformal Vertex Operator W Field Algebra Lie conformal / Current Λ-bracket • Vertex Operator Algebra [Borcherds '86; Frenkel, Lepowsky, Meurman '88; Kac '98; Frenkel, Ben-Zvi '01] M.-A. Fiset 1809.06376 Operator algebras Formalism TCS Examples Operator Algebra [Thielemans '95 hep-th/9506159] Mathematica package 1 Vector space V of operators denoted f1; A; B;:::g. We think of elements of V as operators acting on the Hilbert space of the QFT. b f b 2 Z2-grading V = V ⊕ V and 1 2 V . 3 Even linear map @ : V ! V (e.g. @z ). 4 Sequence of bilinear pairings ·· : V ⊗ V V, n 2 , compatible with n ! Z the grading and s.t. (terms in OPEs) (0) 8 A; B 2 V, 9 n(A; B) 2 such that AB = 0 : Z n≥n(A;B) (1)( unity): 1A = δ A 8A 2 V n 0;n (2)( commutativity): m jAjjBj X (−1) (m−n) BA = (−1) @ AB 8n 2 Z n (m − n)! m m≥n (3)( associativity): X n − 1 A BC = (−1)jAjjBj B AC + AB C m n l − 1 l n m l≥1 m+n−l Normal ordering: :AB: := AB 0 M.-A. Fiset 1809.06376 Operator algebras Formalism TCS Examples Calabi-Yau: Virasoro N = 2 (Md ; g) j Ricci(g) ≈ 0 2-dim N = 1 SCFT N =1 Vir c := G 3 ; T(2) j OPEs 2 \ N =2 ~ Vir c := G 3 ; T(2) ; J(1) ; G 3 ) j OPEs 2 2 Properties / applications: • Spacetime supersymmetry • Spectral flow: Isomorphism NS / R • Chiral ring of special NS fields (chiral primaries) related to Ramond ground states and Dolbeault cohomology • Topological twists • etc. [Lerche, Vafa, Warner '89; Banks, Dixon, Friedan, Martinec '88, etc.] M.-A. Fiset 1809.06376 N =4 4 • e.g.